reasonably efficient enumerator for https://oeis.org/A033178

sumprod.c

#include <stdio.h>
#include <stdlib.h>

int recur(int n, int m, int k, int prevsum, int prevprod) {
    if (m <= 0 || prevprod >= n) return 0;
    int sum = prevsum + k, prod = prevprod * k, rem = (sum + m - 1) - prod;
    int count = (rem >= 0 && prevprod > 1 && rem % (prevprod - 1) == 0);
    for (; sum <= 2 * n && prod < n; ++k, ++sum, prod += prevprod) count += recur(n, m - 1, k, sum, prod);
    return count;
}

int a(int n) {
    if (n <= 0) return 0;
    if (n == 1) return -1;
    return recur(n, n, 2, 0, 1) + (n == 2);
}

int main(int argc, char **argv) {
    if (argc < 2) return 1;
    int limit = atoi(argv[1]);
    for (int i = 2; i <= limit; ++i) printf("%d %d\n", i, a(i));
    return 0;
}